Tesi di LAUREA SPECIALISTICA Titolo Valutazione di opzioni swing tramite metodo Fourier Space Time-Stepping nell ambito di modelli di mercato con salti Data 2010-12-20 Autore/i Testa, Enrico Relatore Sgarra, C. Relatore Marazzina, D. Full text non disponibile Abstract This work was done to show the validity and the potential of the Fourier Space Time-stepping method. During the last decade research for better and faster methods dedicated to option pricing has grown more and more. It would be considered ideal to find a method able to price every kind of existent contingent claim and whose evaluation is fast and precise. A famous example is the algorithm invented by Carr and Madan in 1999. The algorithm allows to price European Options and it’s based on exploiting the characteristic function of the process driving the underlying price process and the Fourier Transform used in its most efficient formulation, the Fast Fourier Transform. The use of FFT allows to drastically reduce the number of flops needed to compute the transform even though in order for it to work it’s needed that the dimension of the vector that has to be transformed is a power of two. Despite the great success of the method, that revealed it self to be extremely efficient and reliable, it is limited to the pricing of European Options and is hardly extendable to pricing of contracts with non standard payoffs. The Fourier Space Time-stepping method can instead be used to price derivatives with non standard payoffs and has great stability and convergence properties. FST valuations will be carried out in jump market models, in particular it’s hypothesized that the underlying assets dynamics are driven by a Lévy process. Lévy processes are generalizations of more standard processes such as the Geometric Brownian Motion and are able to adapt better to historical series of stock prices. Finite and infinite activity Lévy processes will be addressed separately in order to be subsequently compared. The work starts with a chapter dedicated to basic notions needed to understand the fundamentals of Lévy markets and of the Carr-Madan algorithm which is the base from which the Fourier Space Time-stepping algorithm can be understood. The chapter will start with a brief explanation of Lévy processes, their properties and on how can they be build and modeled. Some examples will follow on sample Lévy processes. The chapter will then go on in showing the Fourier Transform, concentrating on the FFT algorithm especially, fundamental in the FST framework. A part on option pricing using characteristic functions will then follow with an extensive explanation on the Carr-Madan formulation. At last a small introduction to pseudo differential operators is proposed, these operators are needed during the solution of the FST method. In chapter 3 Fourier Space Time-stepping will be broadly and extensively explained, from the PIDE solution via Fourier transform that will make it equivalent to an ODE family parameterized by w to appropriate grid selection and stability and convergence properties. Prices will then be computed on different kind of derivatives. European style derivatives will be addressed first exploiting comparison with the Carr-Madan prices. Pricing will be carried out under different kinds of Lévy models and it will be done also a comparison between finite and infinite activity Lévy models. Path-dependent option pricing follows. Barrier options will be priced by making a minor change in the equation that lead to European style option prices, the only difference is in fact the boundary condition. Last but absolutely not least is the original contribution of the work, the pricing of swing contracts. Because of their complexity it will be shown how it will be necessary to introduce a dynamic programming equation in the FST algorithm to handle the problem. In this case also pricing will be done with different kinds of Lévy models, both finite and infinite activity as usual. As last experiment a price comparison will be done using as reference an algorithm that uses forests of trinomial trees.